FOM: comments on Wilson's dual view of foundations

Stephen G Simpson simpson at
Tue Mar 7 15:28:48 EST 2000

Thomas Forster 3 Mar 2000 writes:

 > True NF seems to be interpretationally weak, but this doesn't
 > necessarily hold any moral for the view of set theory that it
 > represents.  It might just not represent it very well.

This is a good point.  Maybe we need to get clearer on exactly what
view of sets gives rise to NF.  Then perhaps we will be able to think
of better formal representions of that view, which might turn out to
be interpretationally richer than NF.

So, what is the view of sets reflected (perhaps imperfectly) by NF?
Can this view be described discursively, i.e., in words?  Is it a
conceptually coherent view?  Is it as coherent as, say, the view of
sets that is reflected (perhaps imperfectly) by ZFC?

The ZFC view of sets seems reasonably coherent and has been described
discursively many times, in terms of the well-known cumulative
hierarchy, obtained by transfinite iteration of the power set
operation along the ordinal numbers.  This is in papers by Zermelo,
Fraenkel, von Neumann, Shoenfield, and others.  In addition, the
cumulative hierarchy has interesting categoricity properties which
tend to increase our confidence in the underlying picture.

Is there a discursive description of the NF view of sets that that is
comparably convincing and/or of comparable clarity?  Are there papers
(by Quine and Rosser perhaps) which provide such a description?

-- Steve

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